Completeness in $L^1 (\mathbb R)$ of discrete translates

Abstract

We characterize, in terms of the Beurling-Malliavin density, the discrete spectra $\Lambda\subset\mathbb R$ for which a generator exists, that is a function $\varphi\in L^1(\mathbb R)$ such that its $\Lambda$-translates $\varphi(x-\lambda), \lambda\in\Lambda$, span $L^1(\mathbb R)$. It is shown that these spectra coincide with the uniqueness sets for certain analytic classes. We also present examples of discrete spectra $\Lambda\subset\mathbb R$ which do not admit a single generator while they admit a pair of generators.